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NUMISMATIC 
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By  DAVID  EUGENE  SMITH,  LL.D 


THE  AMERICAN  NUMISMATIC  SOCIETY 
BROADWAY  AT  156th  STREET 
NEW  YORK 
1921 


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NUMISMATIC 

NOTES  & MONOGRAPHS 


Numismatic  Notes  and  Monographs  is 
devoted  to  essays  and  treatises  on  subjects 
relating  to  coins,  paper  money,  medals  and 
decorations,  and  is  uniform  with  Hispanic 
Notes  and  Monographs  published  by  the 
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Foundation. 


Digitized  by  the  Internet  Archive 
in  2016  with  funding  from 
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https://archive.org/details/computingjetons09smit 


The  Salamis  Abacus 


Found  on  the  Island  of  Salamis  in  1846 


COMPUTING  JETONS 


THE  AMERICAN  NUMISMATIC  SOCIETY 
BROADWAY  AT  156th  STREET 
NEW  YORK 
1921 


COPYRIGHT  1921  BY 
THE  AMERICAN  NUMISMATIC  SOCIETY 


Press  of  The  Lent  & Graff  Co.,  New  York 


THE  GETTY  CENTER 
LIBRARY 


PREFACE 


This  monograph  is  based  upon  an  ad- 
dress delivered  by  the  author  before  the 
American  Numismatic  Society,  in  New 
York  City,  on  February  7,  1921.  The  pur- 
pose is  set  forth  in  the  monograph  itself, 
but  the  author  wishes  to  take  advantage  of 
a prefatory  page  to  express  his  apprecia- 
tion of  the  kindness  of  the  officers  of  the 
Society  in  asking  him  to  prepare  the  ad- 
dress for  publication.  He  also  wishes  to 
acknowledge  the  courtesy  of  George  A. 
Plimpton,  Esq.,  of  New  York  City,  in 
generously  permitting  the  use  of  his  large 
library  of  rare  textbooks  for  the  purpose  of 
preparing  most  of  the  illustrations  used 
in  this  work;  and  to  express  his  thanks  to 
L.  Leland  Locke,  Esq.,  of  Brooklyn, — him- 
self a contributor  to  the  history  of  nota- 
tion and  of  mechanical  computation,  par- 
ticularly in  relation  to  the  quipu, — for  the 
kind  assistance  rendered  by  him  in  taking 
the  photographs. 


1 

COMPUTING  JETONS 
By  David  Eucene  Smith,  LL.D. 

GENERAL  PURPOSE  OF  THE  ADDRESS 

In  accepting  the  invitation  of  the  Ameri- 
can Numismatic  Society  to  speak  upon  the 
subject  of  Computing  Jetons,  I have  nat- 
urally considered  the  possibility  of  offer- 
ing something  that  might  appeal  to  its 
members  as  not  already  familiar.  Few 
works  upon  any  subject  relating  to  numis- 
matics are  so  exhaustive  in  their  special 
fields  as  the  monumental  and  scholarly 
treatise  of  Professor  Francis  Pierrepont 
Barnard  ( Casting-Counter  and  Counting- 
Board,  Oxford,  1916),  and  hence  it  may 
seem  quite  superfluous,  and  indeed  per- 
sumptuous,  to  attempt  to  supplement  such 
a storehouse  of  information. 

NUMISMATIC  NOTES 

2 

COMPUTING  JETONS 

Professor  Barnard,  however,  approached 
the  subject  primarily  from  the  standpoint 
of  a numismatist,  a field  in  which  he  is  an 
acknowledged  expert,  as  witness  the  honor 
that  has  recently  come  to  him  in  his  ap- 
pointment as  curator  of  coins  and  medals 
in  the  Ashmolean  Museum  at  Oxford,  and 
so  it  has  seemed  to  me  that  I might  make 
at  least  a slight  contribution  by  approach- 
ing it  from  the  standpoint  of  a student  of 
the  history  of  mathematics.  It  would,  in 
that  case,  be  natural  to  consider  primarily 
the  need  for,  the  use  of,  and  the  historical 
development  of  the  jeton  in  performing 
mathematical  calculations,  and  this  is  the 
pleasant  task  that  I have  set  for  myself  in 
preparing  this  monograph. 

Although  Professor  Barnard  has  also 
considered  this  field,  I hope  to  contribute 
something  in  the  way  of  illustrative  mate- 
rial, at  least,  and  perhaps  to  make  some- 
what more  prominent  the  early  history  of 
a device  which,  in  one  form  or  another, 
seems  to  have  dominated  practical  cal- 
culation during  a good  part  of  the  period  of 
human  industry. 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

3 

NECESSITY  FOR  AIDS  IN  COMPUTATION 

The  numeral  systems  of  the  ancients 
were  never  perfected  sufficiently  to  allow 
for  ease  in  general  computation.  The  Baby 
Ionian  notation,  adapted  to  a combination 
of  the  numerical  scales  of  ten  and  sixty, 
and  limited  by  the  paucity  of  basal  forms 
imposed  by  the  cuneiform  characters,  was 
ill  suited  to  calculation;  the  Egyptian  and 
Roman  systems  were  an  improvement  but 
they  failed  to  meet  the  needs  of  computers 
when  the  operations  extended  beyond  sub- 
traction; the  several  Greek  systems  finally 
developed  into  something  that  was  rather 
better  than  their  predecessors,  but  they 
also  failed  when  such  an  operation  as 
division  had  to  be  performed  with  what 
we  would  call  reasonable  speed.  The  diffi- 
culty may  easily  be  seen  by  considering 
two  numbers  (6469  and  2399)  written  in 
one  form  of  Roman  notation  of  the  time  of 
the  Caesars: 

vi  00  cccclx  vim 

11  00  ccc  lxxxxviiii 

AND  MONOGRAPHS 

4 

COMPUTING  JETONS 

For  purposes  of  adding,  these  forms  are 
simple  enough.  While  they  take  longer 
to  write  than  ours,  the  actual  addition 
can  be  quite  as  readily  performed  as  by 
us,  and  moreover  it  is  evident  that  no 
addition  table  need  be  learned,  the  entire 
operation  reducing  to  little  more  than 
counting.  When  we  come  to  multipli- 
cation or  division,  however,  the  Roman 
notation  was,  like  practically  all  others  of 
ancient  times,  very  cumbersome.  Even 
as  perfected,  or  at  least  as  changed,  in 
medieval  times,  the  multiplying  of  c.  lxiiij. 
ccc.  1.  i by  .vi.  dc  lxvi  (to  take  two  cases 
from  the  twelfth  century),  or  of  cId.  Id.  ic 
by  Dcccxcj  Uccxxxiiij  q°s  D1  x U (to 
take  a Dutch  form  and  a Spanish  form, 
both  of  the  sixteenth  century)  would  have 
discouraged  almost  any  computer.  Even 
the  greatest  mathematicians  of  antiquity, 
the  Greeks,  had  serious  difficulty  in  using 
their  most  highly  developed  numerals  in 
the  division  of,  for  example,  /ATMB  by 
PE  (that  is,  1342  by  105). 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

5 

There  is  another  reason  why  the  an- 
cient systems  were  such  as  to  demand 
some  kind  of  mechanical  devices  to  aid 
the  computer.  Even  had  our  present  con 
venient  numerals  been  known,  the  an- 
cients had  no  simple  way  of  using  them. 
We  do  our  computation  on  paper,  but  rag 
paper  was  unknown  before  the  first  cen- 
tury, and  our  cheap  paper  is  a very  recent 
invention.  Papyrus  seems  to  have  been 
generally  unknown  in  Greece  before  the 
seventh  century  B.  C.,  although  it  had 
long  been  used  in  Egypt;  parchment  was 
an  invention  of  the  fifth  century  B.  C.; 
while  tablets  of  clay  or  wax  were  quite 
unsuited  to  extensive  numerical  work.  The 
situation  was,  therefore,  a serious  one  for 
those  who,  in  Babylonia,  computed  nu- 
merical tables  for  the  astrologers  and 
astronomers;  and  for  the  merchants  and 
money  changers  of  the  Mediterranean 
countries  who,  after  coinage  appeared  in 
the  seventh  century  B.C.  had  need  of  more 
extensive  calculations  than  their  prede- 
cessors in  the  commercial  field  had  re- 
quired. 

AND  MONOGRAPHS 

6 

COMPUTING  JETONS 

THE  DUST  ABACUS 

To  meet  the  needs  imposed  by  these 
cumbersome  systems  of  notation  the  world 
devised,  from  time  to  time  and  in  differ- 
ent parts  of  the  earth,  various  forms  of 
an  abacus.  Originally  the  term  seems  to 
have  been  used  to  mean  a board  covered 
with  a thin  coat  of  dust  (Semitic  abq, 
dust).  Upon  this  board  it  was  possible  to 
write  with  a stylus,  and  the  figures  could 
easily  be  erased.  Such  devices,  occasion- 
ally referred  to  by  early  writers,  could 
hardly  have  been  of  much  service  except 
in  connection  with  such  temporary  work 
as  the  computation  with  small  numbers. 
Indeed,  among  the  several  doubtful  ety- 
mologies of  the  word  that  have  been  sug- 
gested is  the  one  that  the  Greek  abax 
came  from  alpha  (the  letter  standing  for 
i),  beta  (the  letter  standing  for  2),  and 
axia  (relating  to  value).  The  dust  abacus 
may  also  have  given  the  name  to  the 
gobar  (dust)  numerals,  which  were  used 
by  the  Moslems  in  Spain.  The  instrument, 
therefore,  served  the  same  purpose  as  the 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

7 

wax  tablet  of  the  Greeks  and  Romans 
(a  device  that  remained  in  use  in  Europe 
until  the  eighteenth  century),  as  the  more 
modern  slate,  and  as  the  paper  pad  of  the 
present  day.  The  blackboard  found  in 
our  schools  is  a late  descendant  of  this 
type  of  abacus,  as  is  also  the  wooden  tablet 
used  in  the  native  Arab  schools  at  the 
present  time. 

EARLY  FORMS  OF  THE  LINE  ABACUS 

The  dust  abacus  was  a crude  affair 
compared  with  its  successor,  the  line 
abacus.  This  instrument  had  various 

forms.  At  first  it  seems  to  have  been  a 
ruled  table  similar  to  the  specimen  found 
in  1846  on  the  island  of  Salamis.  Upon  the 
ruled  lines  the  computer  placed  counters 
(Greek  ^ 7<J>o c,  pebbles), — the  units  on  one 
line,  the  tens  on  the  next,  and  so  on. 
Such  instruments  are  referred  to  by  sev- 
eral early  writers,  and  Herodotus,  for 
example,  compares  the  Greek  and  the 
Egyptian  forms,  saying  that  the  inhabi- 
tants of  the  Nile  valley  “write  their  char- 
acters and  reckon  with  pebbles,  bringing 

AND  MONOGRAPHS 

8 

COMPUTING  JETONS 

the  hand  from  right  to  left,  while  the 
Greeks  go  from  left  to  right,”  these  being 
the  respective  directions  taken  in  the 
Egyptian  and  the  late  Greek  writing. 

Sometimes  the  counters  were  placed 
loosely  on  the  lines,  and  sometimes,  though 
at  a much  later  period,  they  were  fastened 
to  the  table  by  being  fixed  in  grooves  or 
by  being  strung  on  wires  or  rods.  Several 
apparently  late  Roman  pieces  showing  the 
grooved  abacus  are  extant,  while  the 
Chinese  swan  pan  shows  the  counters 
strung  like  beads  upon  wires  or  rods. 

THE  ROMAN  COUNTERS 

There  are  numerous  classical  references 
to  the  abacus,  and  particularly  to  the  loose 
counters  from  which  the  later  jetons  were 
derived.  Horace,  for  example,  speaks  of 
the  schoolboy  with  his  bag  and  tablet 
hung  upon  his  left  arm,  the  tablet  being 
some  type  of  abacus,  perhaps  the  one 
covered  with  wax.  Juvenal  mentions  both 
the  tablet  and  the  counters,  and  Cicero 
and  Lucilius  refer  to  brass  counters  when 
they  speak  of  the  aera. 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

9 

The  common  Roman  name  for  these 
counters  was  calculi  or  abaculi.  The  word 
calculus  is  a diminutive  of  calx,  meaning 
a piece  of  limestone  and  being  the  root 
from  which  we  have  our  word  “chalk.” 
A calculus  is,  therefore,  simply  what  we 
call  a “marble”  when  referring  to  a small 
sphere  like  those  which  children  use  in 
playing  games.  From  the  fact  that  these 
calculi  were  used  in  numerical  work  we 
have  the  word  calculare  (literally  “to 
pebble,”  or  “marble”),  meaning  to  cal- 
culate or  compute.  The  word  calculus , 
used  in  this  sense,  was  transmitted  by  the 
Romans  to  medieval  Europe  and  was  in 
common  use  until  the  sixteenth  century. 
When  it  was  abandoned  as  referring  to  a 
counter  it  was  adopted  as  a convenient 
term  to  indicate  the  branch  of  higher 
analysis  which  is  now  generally  known 
as  “the  calculus.”  It  is  still  used  in 
various  languages,  however,  to  refer  to 
elementary  work  with  numbers. 

As  to  the  actual  calculi  used  by  the 
Romans,  we  have  no  specimens  that  can 
be  positively  identified.  Thousands  of 

AND  MONOGRAPHS 

IO 

COMPUTING  JETONS 

small  disks  have,  however,  come  down  to 
us,  generally  classified  as  gaming  pieces, 
and  there  seems  to  be  no  doubt  that  these 
also  served  the  purpose  of  counters.  The 
Romans  have  left  records  of  such  games 
as  the  Ludus  latruncidorum  and  Ludus  duo- 
decim  scriptorum , in  which  they  employed 
pieces  which  they  spoke  of  as  calculi , so 
that  the  disks  that  were  used  in  ancient 
games  like  checkers  and  backgammon 
were  called  by  the  same  name  as  the  com- 
puting pieces.  Indeed,  this  same  custom 
is  found  in  the  case  of  the  jetons  of  modern 
times,  particularly  in  the  seventeenth  and 
eighteenth  centuries,  when  the  computing 
pieces  began  to  be  used  solely  in  gaming, 
a custom  to  which  we  owe  our  poker  chips, 
just  as  we  owe  our  billiard  markers  to  a 
late  form  of  the  Roman  abacus.  It  is, 
therefore,  quite  safe  to  say  that  the  small 
disks  so  often  found  in  Roman  remains 
represent  both  computing  and  gaming 
jetons.  Indeed,  it  is  probable  that  the 
tradesman  paid  little  attention  to  the  size, 
shape,  or  material  of  the  calculi  which  he 
used  in  his  computations. 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

T I 

THE  ABACUS  IN  THE  ORIENT 

The  early  Chinese,  not  only  before  the 
Christian  era  but  for  more  than  a thousand 
years  after  this  era  began,  made  use  of 
counting  rods.  These  were  laid  upon  a 
computing  table  and  were  used  in  some- 
what the  same  way  that  the  jetons  were 
used  in  Europe.  The  rods  were  commonly 
made  of  bamboo,  although  sometimes,  as 
in  the  sixth  century,  iron  pieces  were  used. 
The  early  literature  shows  that  the  wealthy 
class  often  employed  ivory  rods. 

At  least  as  early  as  the  twelfth  century, 
and  we  have  no  positive  knowledge  of 
the  matter  before  that  time,  the  Chinese 
computers  replaxed  the  “bamboo  rods”  by 
sliding  beads,  the  new  instrument  being 
known  as  the  swan  pan  (computing  tray). 
Where  they  obtained  their  idea  we  do  not 
know,  but  there  is  some  reason  for  be- 
lieving that  it  came  from  Central  or  West- 
ern Asia.  At  any  rate  they  adopted  a 
form  that  was  quite  like  the  late  Roman 
abacus  except  that  the  beads  were  made 
to  slide  upon  rods  instead  of  in  grooves. 

AND  MONOGRAPHS 

±JS. 


± - & % « .ft 


• 

- 

1 

III 

1 

I 

4 

% 

-1 

f 

-II 

11 

1 

r 

II 

ll 

— 

\3 

Counting  Rods 

As  shown  in  early  Chinese  works,  being  used  in  this 
case  to  represent  numerical  coefficients  in  algebra 


SlV  1 


COMPUTING  JETONS 

13 

This  form  has  not  changed  materially  since 
the  earliest  illustrations  that  have  come 
down  to  us  in  books  or  manuscripts,  and 
is  still  used  by  all  Chinese  computers  at 
home  and  abroad.  Unless  they,  in  time, 
adopt  some  more  modern  form  of  a calcu- 
lating machine,  there  seems  to  be  no  good 
reason  for  abandoning  the  swan  pan, 
since  it  permits  of  more  rapid  calculation 
than  is  possible  with  pencil  and  paper, — 
at  least  in  the  most  common  numerical 
operations  of  commercial  life. 

In  the  field  of  algebra,  where  the  co- 
efficients that  enter  into  an  equation  are 
usually  relatively  small,  the  rods  con- 
tinued to  be  used  until  European  mathe- 
matics replaced  the  Oriental,  largely  owing 
to  the  influence  of  Jesuit  scholars  in  the 
seventeenth  and  eighteenth  centuries. 

The  Koreans  received  their  mathematics 
from  China  and  transmitted  it  to  Japan. 
The  computing  rods  (their  ka-tji-san)  were 
adopted,  and  they  were  transmitted  to 
Japan  in  the  form  of  chikusaku  (bamboo 
rods),  but  they  were  later  modified  into 
rectangular  pieces  known  as  sanchu  or 

NUMISMATIC  NOTES 
S 

14 

COMPUTING  JETONS 

sangi.  The  rods  remained  in  use  in  Korea 
until  the  nineteenth  century,  and  in  alge- 
braic work  they  continued  to  be  employed 
by  Japanese  scholars  until  the  European 
mathematics  replaced  the  ancient  wasan 
(native  mathematics). 

In  the  sixteenth  century,  however, 
Japan  adopted  a form  of  the  Chinese  swan 
pan , under  the  name  soroban,  improving 
upon  the  shape  and  arrangement  of  coun- 
ters, and  this  instrument  is  still  in  uni- 
versal use  by  her  computers. 

In  Central  and  Western  Asia,  perhaps 
in  the  late  Middle  Ages,  a type  of  abacus 
developed,  which  the  Turks  now  call  the 
coulba  and  the  Armenians  the  choreb. 
It  passed  thence  to  Russia  where  it  is 
known  as  the  stchoty  and  is  still  generally 
used.  The  form  differs  materially  from 
the  Roman  and  Oriental  ones,  but  served 
the  same  purposes.  Each  line  of  this 
abacus  consists  of  ten  beads,  these  being 
strung  on  wires  and  being  so  colored  as  to 
allow  the  eye  to  recognize  without  difficulty 
the  various  groups  of  fives  as  they  appear 
in  the  rows. 

NUMISMATIC  NOTES 

COMPUTING  JETONS 


THE  GERBERT  ABACUS  AND  JETONS 

From  the  standpoint  of  jetons,  only  two 
forms  of  the  abacus,  as  it  appeared  in 
Western  Europe,  have  any  interest  for  us. 
One  of  these  was  called  by  the  early  writers 
the  Pythagorean  Table  ( mensa  Pytha- 
gorica ),  a term  also  applied  to  one  form  of 
the  multiplication  table.  The  other  was 
known  as  the  arc  abacus,  or  Pythagorean  Arc 
{arcus  Pythagoreus ),  but  may  very  likely 
have  been  due  to  Gerbert  (Pope  Sylvester 
II,  c.  1000),  who  is  known  to  have  used  it. 
This  arc  abacus  consisted  of  a table  marked 
off  in  columns  surmounted  by  arcs,  thus: 


The  letters  H,  T,  and  U stand  for 
hundreds,  tens,  and  units. 


AND  MONOGRAPHS 


i6 


COMPUTING  JETONS 


Gerbert  had  an  artisan  make  nine  sets  of 
counters,  and  upon  each  of  the  first  was  the 
figure  i,  upon  each  of  the  second  the  figure 
2,  and  so  on,  those  of  the  last  set  having  the 
figure  9.  If  he  wished  to  represent  the 
number  207,  for  example,  he  placed  the 
counters  as  follows: 


It  will  be  seen  that,  had  Gerbert  known 
the  zero,  he  would  not  have  needed 
counters  at  all,  for  he  would  have  written 
207  on  a wax  tablet.  Since  the  zero  came 
to  be  known  in  Europe  at  about  that  time, 
Gerbert’s  form  of  the  abacus  and  his 
peculiar  jetons  with  numerals  upon  them 
were  very  short  lived,  and  they  made  no 
impression  upon  the  methods  of  mechani- 
cal calculation  employed  by  his  successors. 


NUMISMATIC  NOTES 


COMPUTING  JETONS 

I7 

THE  LATE  EUROPEAN  LINE  ABACUS 

We  are  quite  ignorant  as  to  the  forms 
which  the  abacus  assumed  in  various  parts 
of  Europe  between  the  time  of  the  Fall  of 
Rome,  in  the  fifth  century,  and  the  advent 
of  the  line  abacus  of  the  late  Middle  Ages. 
We  only  know  that  the  earliest  form  that 
has  come  down  to  us  in  the  medieval 
manuscripts  and  in  the  arithmetics  of  the 
first  two  centuries  of  printing  is  substanti- 
ally as  shown  on  pages  18-24. 

The  earliest  printed  illustration  of  the 
arrangement  of  counters  on  the  table  is  the 
one  given  in  the  Algor ithmus  Linealis 
(Leipzig,  c.  1488;  the  facsimile  is  from  the 
edition  of  c.  1490).  The  work  was  pub- 
lished anonymously,  but  was  probably 
written  by  Johann  Widman,  a mathemati- 
cian of  considerable  prominence  and  then 
residing  in  Leipzig.  The  arrangement  of  the 
counters  representing  1,759,876  is  shown  in 
the  column  at  the  right.  The  middle  column 
shows  the  same  number  written  in  a dif- 
ferent fashion  for  purposes  of  subtraction. 
The  column  at  the  left  shows  the  number 

AND  MONOGRAPHS 

^tKiOOOCOO^ 

focooo  o 


4 3 n quttectam  ccrce  pomnoc  Utter*  nonep  Un< 
Gcumjpactotiiq}  pcaeionem  erptimmcee , qaatont 
v(cnisetnamer<tlt6ftgitiftc«ttotn^t6(equenttbu0 

tttttgttoemetiia. 

3 monoe.  v.  quittoe.jt.&frtoa.  topUtvfgmoe 
3C  A • toplat  tbcm . triplet . I r .1  quacp  (olt 
^utnqoagtntafacit . |eb  nonagSntflbat.  re. 
C.brttccrJtcnos.quabJingctjMqooqj  cb. 

5J> . qoocp  quingenta  ft  non  foecitfoctura 


Algorithmus  Linealis 

Probably  by  Johann  Widman,  c 1468 


y#he  bur&  bit  siffcr  Iemdfotfifrcn  mb  anfo 
{piety? . 0inrcil  id>  ini  anf<m0  bid  bunion* 
i>oe|dbi0  tfenugfam  of  Itret/tril  i<b  anfabm 
port  bcbcumud  ber  linien  alfo  # 0ic  pm trifle  U* 
ni  bebcut  tind/bic  anber  ob  j r $el)c/  bie  b:it  bun 
bert/bie  vierb  raufent/bic  f&nflt  jcbcntaiifcnt  / 
flifo  fo:t  bie  ne0ft  pberfieb  atoeg  jcbcnmi  fouif, 
©jejklbmi#  $tt»if<bm  $wcyc  linicn  tycifi  bad  fp« 
cinm/ctiltbslbfoml  aid  bit  »c0(l  ober  Imi/  ober 
(unflniaTiouilab  bic  negff  pnrer  lini/wicnacb* 
uolgenbe  (i0nr  tl>ut  abrreyferj. 


,£ffnjf&m/-raur __  ;oooo<j 

^unbert  tan:  I 100000 

Jirnffeigwu: } foooo 

^ebchcao:  I 10000 

Jriinfftaii:  * 5000 

gaufent  1000 

/ur.ffpunbcrt^ foo 

^imbcrt  100 

Jun(fei0  fit 

$cbcn  *0 

SanfH  r 

£in&  I 

£inbalbd  i 

In  aufpre&mftciner  $al  / hcbobttianfmfcb 
to  d bunbert  famytfewe  obern  fPAdo  tfpein  aud 
fonft oim  glared  wo  linim  fampt  ire  obem  fpy 
ricfy^famm  5a!  ^ * 

von  meb*  uregav  fol  to* 


Christoff  Rudolff’s  Kunstliche  rcchnung  mil 
der  zijfer  vnnd  mit  den  zalpfennige , 1526 

From  the  edition  of  1534 


20 

COMPUTING  JETONS 

1,666,666.  A small  cross  was  usually 
placed  upon  thousands’  line,  and  one  on 
millions’  line,  as  here  shown,  the  purpose 
being  to  aid  the  eye  in  reading  the  numbers. 

Although  there  were  special  modifica- 
tions of  the  line  abacus,  the  general  type 
is  the  one  on  page  19.  The  illustration  is 
from  Christoff  Rudolff’s  Kunstliche  rech- 
nung  mit  der  zifer  vnnd  mil  den  zalpfen- 
ninge  (Vienna  or  Niirnberg,  1526;  the 
facsimile  is  from  the  Niirnberg  edition  of 
1534,  fol.  D.  vj.  v).  Rudolff  was  one  of  the 
best  German  mathematicians  of  his  time, 
and  counter  reckoning  made  very  little 
appeal  to  him.  Nevertheless,  in  writing  an 
arithmetic  for  popular  use,  he  was  forced 
to  include  it.  He  gave  only  one  illustration 
of  the  counting  board,  as  here  shown,  but 
he  explained  the  use  of  the  device  in  per- 
forming the  several  elementary  operations 
as  they  were  reached  in  the  text. 

An  interesting  variant  of  the  table  given 
by  Rudolff  is  one  here  shown  from  the 
Arithmetica  of  a Polish  teacher,  Girjka 

e e 

Gorla  z Gorrlssteyna,  whose  book  appeared 

NUMISMATIC  NOTES 

t*  weffe  pocubtj  ge|t  pcce  u vUtwbe/ttty 
tyobCJeimy  &mye/  gw}  gefJ  ^rqjfcni 
$nA\mmttd/  3«ow4  poctj  My  Qttcn  Zifyc/ 
pat*™  H qms  iDefct  Ziffc  / ©feffcw  iltj'm I 
Bw  rifyc/  QcbmaiilLjmj  Cifyc  Cifycftw. 

tfSpafc  Spacium  ne$  pole  me$y  tUmrni 
& fry  pete  tzicyteltf  pftwj  ytfo  P^ttaftee 
trw  f#&lfgqf  $ UteZthuk  p^umqf*' 


Wyfivhkmi  Jltjn  <?  @ptv 


a(J-  l«  c « O"  c-  0®  >>■  0 — ^ 
<00000 
1 '0'0-0<C*C— * 

c 0 0 0 0 

c?um. 

I* 

£— (Tiffc  CijycS — o> 
pet@>crfcifyc 
0to-Cifyc  — fr> 
|^ai>cfrtt  Cifyc 

— {?>  *» f/* 1 ..7T 

fc  0 0 

- — ^ 

tt«L 

“ |L  *"bC. IJJ/w'*  IJV 

pitCiff'c 

1 m '"Ttr’s.1  V y 

f 00 

\ wi|yt  1 ■ 

pet  ecr 

r 0 

•® 

padefst 

S' 

-A  - - - -« 

pit 

a 

% 1 

pfil 

pnwmob'f inaP/  natMaoMitmu 
ptftft  poI«5i/l«  M toltfo  gebrw  jnmtnd/ 
Spatim  prtitq  p&l/  im$  »j  pft/jt>e»N 


Gorla’s  Arithmetica,  1577 


; 22 

COMPUTING  JETONS 

at  Czerny  in  1577.  This  particular  work 
has  been  selected  partly  because  of  its 
rarity,  and  partly  because  the  form  of  the 
explanatory  diagram  differs  somewhat 
from  the  more  common  type  found  in  other 
parts  of  Europe. 

A further  illustration  of  the  method  of 
explaining  the  table  may  be  seen  from  the 
line  abacus  shown  in  Spanlin’s  Arithmetica 
(Niirnberg,  1566,  page  8). 

When  arranged  for  monetary  computa- 
tion, the  table  was  commonly  divided 
into  columns,  each  being  called  a Banckir 
or  a Cambien.  In  each  Banckir  there  were 
placed  counters  to  represent  respectively 
pounds,  shillings,  and  pence,  or  similar 
denominations  according  to  usage  of  the 
country.  The  illustration  on  page  24  is 
from  Das  new  RecKepuchlein  of  Jakob 
Kobel  (Oppenheim,  1514,  but  from  the 
1518  edition,  fol.  viii,  r).  The  page  has  a 
further  interest  in  the  fact  that  both 
Roman  and  Hindu-Arabic  numerals  are 
shown,  although  in  general  Kobel  preferred 
the  former  as  being  the  ones  more  com- 
monly used  in  his  day. 

NUMISMATIC  NOTES 

s 

ftacm/  unfer  ft  ir  rr  ftm'/  ftm’f 

felb  limcn/rcic  fyte  tm  n>cr<f  bcrait  scft^rn*. 
^rfc  (Eambi,  2lnfrr  tfambi. 

X 

m 
c 
X 


i 

2Cuf?  biftm/roie<$fft6f/  tto^t/foeefkfcb^ 
gr^D^tucn  ^afpfefnnginemifpafm  frae/ 
bifft  Ibc  n ^ffgrljebf  / vnb  finer  frarfur 
<wff  Pie  nr  cf>jt  litii  fyitmff  gclcQt  tcetMer 
#kicf}en  fo  f Safpfcmng  aufi  finer  linicybie 
frffcen  folic  auefy  miffge  bt/nn  finer  twfur 
^ttuiuffin  Das  nccfyft  fpa<w$elc$t  trfrben : 
tnif  tifrcrOf  j)aftf6  9iunac»jc/|a(lu  in  Pc  fpc^ 
cif$  «c  c^jl  t?oI$e  n D <$migfam  ^uuermertf  c* 
2tt>Pifio. 

'Stem/  &ner  $ibt  mifj  $u  *}6ib\in$l  74  ft 
16  g.  $ f dg  / me fj:  3 r n f; p*o  ftymait}/ 

ft  10  fj^p  fcDr  M/$efy  wtfojtf  Oarawf 


wu|fnt 


tauftnt \( 

kunberf 

$c§e  n— — 
tint 


funffiduftnt 

fanffjjun&m 

funff  ^ 
cin  fjalbe. 


Spanlin’s  Arithmetical  1566 


vm 


*Req>enbanc& 

Dw(&#^OitcFu 

obetCambtcn 

IDtc^tDcit^as/ 
ftif  obetCambtcit 

3DfelDtj*}&atfft 

obecCarunbien 

(Bulbcii 

m 

8 

19 er2Swelt?0n&etfcbe?t  iff  do 

^airii^0itu  x>h  Kccbefmma  fo  batuf  gelegt  fern. 

|:f  H^ar  ifr/  Das  Hie  unfierfr  limg/  (Euta  terror. 

^^Wcr5/tTaufaJtr/f  JDte  .f imfft/grclje  'fouifaiit/ 
Store  B<cty!/Dfi6ett  tTaufanr/S"  2Dtc  QvbcM/%w* 
fant  'iraufattt  i c.&nb  alfo  auff  wib  auff  3&$ele/©o  w! 
bet£tiuw  gcmacbt  toe^m/*25ereur  cm  yebe  Ctntg  r>e^ 
mal  ale  \>a  ale  bie  acd#  £mt£  mb’  tt-lDee  *u  ftc!?tt 
lichee  anfc^atDunff  mm  top  (Bpcmpcl 

‘^Eaufantmal  'famfatiHR**7 ®— 

DunberttCaufant C^- — ®- 

=>  c^en  Cawfanr — — ®- 

'rtaufant— — i II 


■1000000-- 
-IOOOOO — 

-10000 

-1000 


Dimka* 

c- — & — 

y?  A 

- — -100 

=>tl?cn 

(£vne 

ti  ijg 

IV 

— lr 

Jakob  Kobel,  Das  new  Rechepuchlein,  1514 

From  the  edition  of  1518 


COMPUTING  JETONS 

25 

Arithmetics  that  related  to  the  use  of 
counters  on  the  line  abacus  were  called  by 
such  names  as  Algor ismus  linealis,  Al- 
gorithmus  linealis , and  Rechenbuchlein  auff 
der  Linien  (Albert,  1534).  The  word 
algorismus  referred  to  arithmetics  that  did 
not  use  counters.  It  is  a medieval  Latin 
form  of  the  Arabic  al-Khowarizmij  that  is, 
“the  man  from  Khwarezm,”  the  country 
about  the  modern  Khiva.  This  man  was 
Mohammed  ibn  Musa  al-Khowarizmi, — 
“Mohammed  the  son  of  Moses,  the 
Khwarezmite,”  the  first  of  the  Arab 
writers,  under  the  Caliphs  at  Bagdad,  to 
prepare  a noteworthy  arithmetic  based 
upon  the  Hindu-Arabic  numerals.  There 
was,  therefore,  no  propriety  in  speaking  of 
a “line  algorismus ,”  since  algorism  was 
quite  the  opposite  of  reckoning  with 
counters  on  the  line  abacus.  The  original 
meaning  of  the  term  was  lost  in  the  late 
Middle  Ages,  however,  and  the  word 
algorismus  was  applied  to  both  types  of 
arithmetic.  Some  of  the  textbooks,  such 
as  the  popular  German  one  by  Adam  Riese 
(1522),  taught  both  counter  and  written 

NUMISMATIC  NOTES 

26 

COMPUTING  JETONS 

reckoning,  and  bore  such  names  as  the  one 
which  this  famous  Rechenmeister  gave  to 
his  second  work,  Rechnung  auff  der  Linien 
vnd  Federn  (Computing  on  the  lines  and 
with  the  pen).  Similarly,  Jodocus  Clichto- 
veus,  a native  of  Nieuport,  in  Flanders, 
published  in  Paris  (c.  1507)  his  Ars 
supputadi  tam  per  calculos  q3  notas  arith- 
meticaSy  a work  which  represented  about 
the  last  of  the  old  counter  reckoning  in  the 
higher  class  of  Latin  arithmetics  published 
in  France. 

A boy  (for  the  girl  rarely  learned  any- 
thing about  computing)  who  knew  the  line 
abacus  was  said  to  “know  the  lines.”  So 
Albert,  who  wrote  in  1534,  says:  “Die 
Linien  zu  erkennen,  ist  zu  mercken,  das 
die  underste  Linien  (welche  die  erste 
genent  wird)  bedeut  uns,  die  ander  hinauff 
zehen,  die  dritte  hundert,”  and  so  on. 
When  he  represented  a number  by  means 
of  counters  on  the  line,  he  was  said  to 
“lay”  the  sum,  as  when  the  same  writer 
says,  “Leg  zum  ersten  die  fl.,”  an  expres- 
sion that  may  be  connected  with  the 
present  one  of  laying  a wager.  He  was 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

27 

often  admonished  to  “lay  and  seize”  care- 
fully, as  in  the  familiar  old  German  distich, 

“Schreib  recht  | leg  recht  | greif!  recbt  J sprich  recht  j 
So  koempt  allzeit  dein  Facit  recht/’ 

in  which  the  term  facit  had  been  brought 
over  from  the  Latin  schools. 

The  intervals  between  the  lines  ( lineae ) 
were  called  “spaces”  ( spatia  or  spacia).  In 
performing  the  operations,  however,  and  in 
representing  different  monetary  units  like 
pounds,  shillings,  and  pence,  it  was  con- 
venient to  divide  the  abacus  vertically,  as 
already  stated.  It  was  because  these  divi- 
sions were  used  particularly  by  the  money 
changers  that  they  were  known  to  the 
German  merchants  not  only  as  Banckir 
but  as  Cambien,  or  Cambiere,  from  the 
Italian  cambia  (exchange), — one  of  many 
illustrations  of  the  indebtedness  of  northern 
merchants  to  their  fellow  tradesmen  and 
bankers  in  the  South.  The  Cambien  were 
also  called  “fields”  ( Feldungen ). 

The  use  of  such  a term  as  Cambien  sug- 
gests the  desirability  of  beginning  the  study 
of  the  line  abacus  in  Italy.  This,  however, 

AND  MONOGRAPHS 

28 

COMPUTING  JETONS 

is  not  a satisfactory  plan,  for  the  Italians 
were,  owing  partly  to  geographical  reasons, 
the  first  of  the  leading  European  nations 
to  adopt,  for  practical  mercantile  purposes, 
the  Hindu-Arabic  numerals,  and  hence 
they  had  generally  abandoned  the  line 
abacus  as  early  as  the  twelfth  century. 
Indeed,  when  Leonardo  Fibonacci  wrote 
his  great  treatise  on  arithmetic,  in  the  year 
1202,  he  felt  justified  in  calling  it  the  Liber 
Abaci  although  the  abacus  is  not  described 
anywhere  in  the  work,  showing  that  the 
term  had  already  come  to  mean  simply 
arithmetic.  We  have  no  treatise  extant 
that  gives  us  any  clear  information  as  to 
how  the  earlier  Italians  of  the  Middle  Ages 
computed  with  the  counters.  By  the  open- 
ing of  the  Renaissance  the  art  was  a lost 
one.  The  Venetian  patrician,  Ermolao 
Barbaro,  who  died  in  1495,  said  that,  in  his 
time,  such  devices  were  used  only  among 
the  barbarians,  having  been  so  long  since 
forgotten  in  Italy  as  to  need  explanation, — 
“Calculos  sive  abaculos  . . . eos  esse 
intelligo  . . . qui  mos  hodie  apud  barbaros 
fere  omnes  servatur.” 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

29 

By  way  of  contrast  with  the  situation  in 
Italy,  Heilbronner,  in  his  Eistoria  Mathe- 
seos  Universae  (Leipzig,  1742)  says  that, 
even  as  late  as  about  the  middle  of  the 
eighteenth  century,  counters  were  used  by 
merchants  in  Germany  and  even  in  France, 
— “in  pluribus  Germaniae  atque  Galliae 
provinciis  a mercatoribus,” — defining  the 
art  of  computing  on  the  line  in  these 
words:  “arithmetica  calculatoria  sive  line- 
aris est  Scientia  numerandi  per  calculos  vel 
nummos  metallicos.” 

The  method  of  using  the  line  abacus 
varied  considerably.  In  rare  cases,  only  the 
lines  were  used,  each  line  counting  as  tens 
of  the  line  just  preceding,  a method  having 
a counterpart  in  the  Russian  stchoty  of  to- 
day. In  others,  only  the  spaces  were 
used,  the  plan  being  similar  to  the  one  just 
mentioned.  Specimens  of  this  type  of 
abacus  are  to  be  seen  in  the  National 
Museum  at  Munich  and  in  the  Historical 
Museum  at  Basel.  In  this  form  the  spaces 
generally  represented  monetary  values, 
such  as  farthings,  pence,  shillings,  pounds, 
10  pounds,  100  pounds,  and  1000  pounds. 

AND  MONOGRAPHS 

30 

COMPUTING  JETONS 

NAMES  FOR  COUNTERS  OR  JETONS 

Since  the  counter  was  cast,  or  thrown, 
upon  the  computing  board,  the  name 
applied  to  it  was  often  connected  with  the 
word  “cast”  or  “throw.”  The  Medieval 
Latin  writers  followed  those  of  classical 
times  in  calling  counters  by  such  names  as 
calculi  and  abaculi,  but  later  computers 
also  recognized  the  notion  of  casting.  On 
this  account  they  gave  to  the  counters  the 
name  projectiles  {pro-,  ahead,  + jacere, 
to  cast).  In  translating  this  term  the 
French  dropped  the  prefix,  leaving  only 
jectiles,  which  they  translated  as  jetons , 
with  such  variations  as  jettons,  gects,  gectz, 
getoers,  getoirs,  jectoirs,  and  gietons.  Refer- 
ring to  the  casting  of  the  counter,  in  con- 
nection with  which  we  still  hear  occasion- 
ally the  expression  to  “cast  up  the  ac- 
count,” the  older  French  jetons  frequently 
bore  such  inscriptions  as  “Gectez,  Enten- 
dez  au  Compte,”  and  “Jettezbien,  quevous 
ne  perdre  Rien.”  Similarly  the  Spanish  com- 
puters spoke  of  the  git  on,  but  they  early 
abandoned  the  use  of  the  abacus. 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

31 

The  Netherland  pieces  were  called  Werp- 
geld,  that  is,  “cast  money”  or  “thrown 
money.  They  were  also  known  by  the  name 
of  Leggelt,  that  is,  “laid  money,”  as  in  pieces 
bearing  the  legend  “Leggelt  van  de  Munters 
van  Holland.” 

In  England  the  common  name  for  the 
computing  disk  was  “counter,”  a word 
which  came  down  from  the  Latin  com- 
piitare  through  such  French  forms  as 
conteor  and  compteur,  appearing  in  Middle 
English  as  counter e,  and  contour.  Thus  we 
are  told,  in  a work  of  the  early  part  of  the 
fourteenth  century,  to  “sitte  doun  and  take 
countures  rounde  . . . And  for  vche  a 
synne  lay  thou  doun  on  Til  thou  thi  synnes 
haue  sought  vp  and  founde,”  a passage 
that  suggests  an  early  use  of  the  rosary,  a 
symbol  found  in  one  form  or  another  in  the 
ceremonies  of  various  religions.  Indeed, 
the  whole  subject  of  bead  counting  or 
fingering,  not  merely  among  Christians 
but  also  among  Buddhists  and  Moham- 
medans, is,  like  knot  tying,  closely  connected 
with  the  abacus,  and  each  has  an  extended 
and  interesting  history. 

AND  MONOGRAPHS 

32 

COMPUTING  JETONS 

In  an  English  work  of  1496,  mention  is 
made  of  “A  nest  of  cowntoures  to  the 
King/’  and  in  the  laws  of  Henry  VIII 
(1540)  there  is  the  expression  “for  euery 
nest  of  compters,”  so  that  the  use  of  “nest” 
to  indicate  the  receptacle  of  the  counters 
was  for  a long  time  common  in  England. 
Such  a nest  may  very  likely  be  referred  to 
by  Barclay  (1570)  when  he  speaks  of  “The 
kitchin  clarke  . . . Jangling  his  counters.” 
When  Robert  Recorde,  the  first  of  the 
noteworthy  writers  upon  mathematics 
whose  works  appeared  in  the  English 
language,  wrote  his  well-known  Ground  of 
Artes  ( c . 1542),  counter  reckoning  had 
begun  to  occupy  a subordinate  place  in  the 
arithmetical  training  of  the  schoolboy. 
Not  until  the  second  part  of  his  book, 
therefore,  does  Recorde  say,  “Nowe  that 
you  haue  learned  the  common  kyndes  of 
Arithmetike  with  the  penne,  you  shall  see 
the  same  arte  in  counters.”  A century 
later,  in  an  edition  of  this  same  popular 
work,  a commentator  speaks  of  ignorant 
people  as  “any  that  can  but  cast  with 
Counters,”  reminding  us  of  Shakespeare’s 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

33 

contemptuous  reference  to  a shopkeeper 
as  being  merely  a “counter  caster.” 

From  the  use  of  the  word  “counter”  in 
the  above  sense  there  came  its  use  to 
designate  an  arithmetician.  An  example  of 
this  is  found  in  a sentence  of  Hoccleve’s 
(1420):  “In  my  purs  so  grete  sommes  be, 
That  there  nys  counter  in  all  cristente 
Whiche  that  kan  at  ony  nombre  sette.” 
The  word  also  came  to  mean  the  abacus 
itself,  as  when  Chaucer,  referring  to  al- 
Khowarizmi  as  Argus,  says: 

“Thogh  Argus  the  noble  covnter 
Sete  to  rekene  in  hys  counter/’ 

From  this  custom  came  the  use  of  the 
word  to  mean  the  table  over  which  goods 
were  sold  in  a shop.  The  expressions 
“counting  house”  and  “counting  room” 
are,  of  course,  of  similar  origin. 

By  reason  of  the  resemblance  of  the 
counter  to  the  common  coins  it  was  often 
called  by  such  names  as  nummus  and 
denarius  projectilis,  somewhat  as  we,  in 
America,  speak  of  a cent  as  a “penny,” 
although  the  two  are  not  the  same  in  value. 

AND  MONOGRAPHS 

34 

COMPUTING  JETONS 

THE  EXCHEQUER 

Although  the  Court  of  the  Exchequer, 
or  the  Chambre  de  Pechiquier,  would 
hardly  seem  to  be  connected  with  the  jeton, 
the  relation  is  an  intimate  one.  The  best 
source  of  our  knowledge  of  this  relationship 
is  the  Dialogus  de  Scaccario  of  one  Fitz- 
Neal,  who  wrote  in  1178.  His  work  is 
written  in  the  form  of  a catechism,  the 
questions  being  proposed  by  a “disciple” 
and  the  answers  being  given  by  the 
“master.”  It  is  written  in  Latin  and  the 
word  scaccarium  is  used  for  exchequer, 
from  the  old  French  eschequier , and  the 
Middle  English  escheker.  Substantially  the 
same  word  was  used  in  Italy  in  the  fif- 
teenth century  to  designate  a plan  of 
multiplication  in  which  the  figures  were 
arranged  as  on  a checkerboard.  This  gave 
rise  to  the  “moltiplicare  per  scacchiero” 
in  the  early  years  of  the  Renaissance 
period. 

In  answer  to  a question  from  the  disciple 
as  to  the  nature  of  the  exchequer,  the 
master  replies: 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

35 

“The  exchequer  is  a quadrangular  sur- 
face about  ten  feet  in  length  and  five  in 
breadth,  placed  before  those  who  sit 
around  it  in  the  manner  of  a table,  and  all 
around  it  there  is  an  edge  about  the  height 
of  one’s  four  fingers,  lest  anything  placed 
upon  it  should  fall  off.  There  is  placed 
over  the  top  of  the  exchequer,  moreover, 
a cloth  bought  at  the  Easter  term,  not  an 
ordinary  one  but  a black  one  marked  with 
stripes  being  distant  from  each  other  the 
space  of  a foot  or  the  breadth  of  a hand. 
In  the  spaces  moreover  are  counters  placed 
according  to  their  values.  . . . Although, 
moreover,  such  a surface  is  called  ex- 
chequer, nevertheless  this  name  is  so 
changed  about  that  the  court  itself,  which 
sits  when  the  exchequer  does,  is  called 
exchequer.  ...  No  truer  reason  occurs  to 
me  at  present  than  that  it  has  a shape 
similar  to  that  of  a chessboard.  . . . The 
calculator  sits  in  the  middle  of  the  side, 
that  he  may  be  visible  to  all,  and  that  his 
busy  hand  may  have  free  course.” 

The  further  description  shows  that,  while 
the  table  was  not  the  ordinary  line  abacus 

AND  MONOGRAPHS 

36 

COMPUTING  JETONS 

already  described,  the  method  of  comput- 
ing was  essentially  the  one  commonly  used 
with  counters.  The  court  itself  was  there- 
fore connected  with  the  royal  treasury 
and  later  with  various  financial  matters 
of  the  realm.  Indeed,  just  before  Fitz-Neal 
wrote  there  appeared  a record  of  “John 
the  Marshal”  being  engaged  “at  the 
quadrangular  table  which,  from  its  coun- 
ters (< calculi ) of  two  colors,  is  commonly 
called  the  exchequer  (scaccarium),  but 
which  is  rather  the  King’s  table  for  white 
money  ( nummis  albicoloribus) , where  also 
are  held  the  King’s  pleas  of  the  Crown.” 
In  this  connection  it  is  interesting  to  recall 
the  fact  that  the  checkered  board  is  still 
quartered  the  arms  of  the  Earl  Marshal  of 
England. 

It  may  be  mentioned,  although  any 
discussion  of  the  subject  at  this  time  would 
carry  us  too  far  afield,  that  the  subject  of 
counters  is  also  connected  with  the  tally 
stick,  with  finger  reckoning,  and  even  with 
the  modern  calculating  machine,  each  of 
which  devices  has  an  extended  and  interest- 
ing history. 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

37 

METHOD  OF  COMPUTING  WITH  JETONS 

Jetons  were  used  for  all  the  elementary 
numerical  processes.  These  generally  in- 
cluded notation,  addition,  subtraction 
doubling,  multiplication,  halving,  division, 
and  roots.  Some  books  had  special  treat- 
ments for  the  Rule  of  Three  and  progres- 
sions. Doubling  and  halving  were  ancient 
processes,  going  back  to  early  Egyptian 
times  and  intended  primarily  to  assist  in 
multiplication,  division,  and  the  treatment 
of  fractions. 

It  will  suffice  to  show  the  general  nature 
of  the  use  of  the  counters  if  we  consider  a 
few  illustrations  from  the  early  printed 
books  and  manuscripts  on  arithmetic.  For 
this  purpose  I have  selected  cases  not 
individually  considered  (with  one  excep- 
tion) in  Professor  Barnard’s  treatise. 

As  to  notation,  this  has  already  been 
sufficiently  explained.  It  will  make  the 
subject  seem  somewhat  more  real,  how- 
ever, if  we  consider  a single  illustration  of 
the  counting  board  laid  for  actual  use. 
Several  such  illustrations  are  given  on  the 

AND  MONOGRAPHS 

Min  Iflav  geo:Onet  *ifUcb 
m btecblm  auf  oenltmeti 
mtt  Tftecben  pfemngcn : oat 
IJangai  angenoeu  511  baf 

licbcm  «ctm»ucb  vno  ben© 
eln  Icvcbtbcbjo  lenten 

mtt  ftgurert  vrtb  ejeeropclit 

VoU$fyema<&tUx* 


Jakob  Kobel,  Ain  Nerv  geordnet  Rechen- 
biechlin,  1514 

Illustrating  the  placing  of  the  counters 


COMPUTING  JETONS 

39 

titlepages  of  sixteenth-century  arithmetics, 
but  one  of  the  clearest  is  found  in  KobePs 
Ain  Nerv  geordnet  Rechenbiechlin  auf  den 
linien  mit  Rechenpfeningen  (Augsburg, 
1514)  and  is  here  shown  in  facsimile.  One 
Cambien  has  the  number  26  and  the  other 
has  485  (with  possibly  one  or  more  counters 
on  the  lowest  line). 

As  to  the  further  operations,  my  only 
excuse  for  venturing  into  a field  which 
Professor  Barnard  has  so  thoroughly 
treated  is  that  this  elementary  presenta- 
tion may  serve  to  popularize  the  subject 
and  that  I may  place  before  those  who  are 
interested  in  computation  certain  fac- 
similes that  are  not  to  be  found  in  his 
treatise.  Professor  Barnard  has  considered 
chiefly  the  works  of  Gregorius  Reisch 
(1503),  Nicholas  von  Cusa  (1514),  Kobel 
(1514),  Sileceus  (1526),  Robert  Recorde 
(c.  1542),  Trenchant  (1566),  Perez  de 
Moya  (1573),  John  Awdeley  (the  printer, 
1574,  the  author  being  unknown),  and 
Francois  Legendre  (1753,  not  the  great 
Legendre),  besides  the  anonymous  Li  Liure 
de  Getz  ( c . 1510).  These  authors  he  has 

NUMISMATIC  NOTES 

40 

COMPUTING  JETONS 

considered  more  fully  than  could  well  be 
attempted  in  the  space  at  my  disposal. 
Since  some  of  the  works  represent  the  best 
sources,  I am  compelled  to  refer  to  them, 
however;  but  in  the  main  I have  given 
illustrations  from  other  sources  in  order  to 
supplement  his  treatment  in  certain  par- 
ticular features. 

The  illustration  from  Caspar  Schleup- 
ner’s  Rechenbuchlein  Auff  der  Linien 
(Leipzig,  1598)  shows  how  the  table  was 
arranged  for  the  reduction  of  Thalers  to 
Groschen  and  Hellers.  The  problem  is  to 
reduce  9 Thalers  to  Groschen  and  then  to 
Hellers,  36  Groschen  being  equal  to  a 
Thaler,  and  108  Hellers  being  equal  to  a 
Groschen.  The  left-hand  Banckir  denotes 
9 Thalers,  the  result  of  the  reduction  to 
Groschen  (324)  appears  in  the  next  column, 
and  the  result  of  the  reduction  to  Hellers 
appears  at  the  right. 

Schleupner  was  one  of  the  last  of  the 
Niirnberg  Rechenmeisters  to  give  serious 
attention  to  counter-reckoning.  The  work 
has  few  equals  in  the  way  of  a simple 
presentation  of  the  subject. 

NUMISMATIC  NOTES 

9?uttf0lg<tt  feeder  3J?dsifciiac&tw 

^£alcr  grcfc&en  9te  fblasfon/ef  * 

Dagegen  n?iDotumfs»  Der  §clfer  minD 
grofcljen/  :f*  Ofcbiweion  auc^ 
gemoefe* 


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vonfomxn  an  / ha),  Dot  9,  ft  gcgen  Dm 
grofc/jen  tntb  fylhm  anftgcfi/fb^aflu  to$ 
Ofrlblmrn  p>kbcr  groffen  uK&ngti  in  flew 
nere/Daj?  Die  9*  cakr  3 *4 *SK/  »nD|ble^c 
3 24*  <£  3 3 SS.^wacjjert/  fa  DuaDcr 
geDarfae  ftgur  wn^en^ttem  an/  *u  rilcf 
anfi^(t/  fo§a|luDagegen  Dae  DfeDuam 
fokbfr  fleimninun^e  in  grbffcrc/  Dafi  Da* 
gegmDurc($  Die  Auction/  Die  3 S 8 8 * $ 
3H8S«n^u  3 24,3? Die  staler  ma» 
cs«i.  0o§n* 


Caspar  Schleupner,  Rechenbuchlein  AujJ  der 
Linien,  1598 


Illustrating  reduction  of  monetary  units 


42 

COMPUTING  JETONS 

The  fundamental  operations  in  arith- 
metic which  we  commonly  limit  to  addi- 
tion, subtraction,  multiplication,  and 
division,  were  subject  to  no  such  narrow 
limitation  in  the  medieval  and  renaissance 
periods.  As  already  stated,  numeration, 
doubling  (duplation),  halving  (mediation), 
roots,  and  certain  other  processes  were 
often  included.  To  illustrate  the  process  of 
doubling,  for  example,  13  times  47  was 
often  found  by  taking  2 times  2 times  2 
times  47,  adding  2 times  2 times  47,  and 
then  adding  47,  thus  reducing  the  process 
to  doubling  and  adding.  In  a somewhat 
similar  way,  the  reverse  operation  can  be 
reduced  to  subtraction  and  halving. 

The  illustration  here  given,  from  Adam 
Riese’s  Rechenbuch  Vjf  Linien  mnd  Ziphren 
(Erfurt,  1522,  but  from  the  Frankfort 
edition  of  1565  fol.  6,  v),  shows  the  reckon- 
ing board  and  a slight  explanation  of  the 
methods  of  doubling,  with  three  examples 
in  our  common  numerals.  Although  Riese 
was  one  of  the  greatest  Rechenmeisters  of 
Germany,  his  explanations  of  the  process 
with  counters  were  not  so  satisfactory  as 

NUMISMATIC  NOTES 

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fcatm  aba#i»fti  tytebemfpacfo/fo  tbuuucgt 
fagt.jDc^fe^cn  mft  oen  b£  miff  ben  ftmen/ 
fo  (4*13  bitf  nid)te  mdyz  -;tt  btipltm  votftrtn- 
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SD&sptdbtralfo  / bafbttbfe  jaf/btefom^ 
mm  ffl  mi£  bem  bttpfrm  / fo  Pomp tbic  ape 
^i*fFgdggt?4limber. 


6ey|jt 

Adam  Riese,  Rechenbuch  Vjf  Linien  vnnd 
Ziphren,  1522 

From  the  edition  of  1565.  Illustrating  doubling 


44 

COMPUTING  JETONS 

those  of  various  other  writers,  as  may  be 
inferred  from  the  case  shown  on  page  43. 

The  illustration  of  halving,  here  given, 
is  from  Johann  Albert’s  Rechenbuchlin  Auff 
der  Federn  (Niirnberg,  1534,  but  from  the 
Wittenberg  edition  of  1561,  fol.  B,  vj,  r ) 
and  has  a much  better  explanation  than 
that  given  by  Riese  in  connection  with 
doubling.  The  problem  is  to  halve  the 
number  3894.  Albert  begins  with  units 
(“grieff  auff  die  vnterste  Linien”)  and 
takes  half  of  4,  which  is  2.  The  rest  of  the 
solution  is  shown  in  the  facsimile,  the 
result  being  1947,  as  set  forth  in  the  right- 
hand  column. 

In  subtracting  one  number  from  another, 
the  counters  were  often  set  down  in  two 
columns  with  a line  between  them,  after 
which  the  subtraction  was  performed  some- 
what as  we  perform  it  now.  A better  plan, 
however,  was  first  to  set  the  larger  number 
down  in  counters,  then  to  write  the  smaller 
number  for  reference,  and  finally  actually 
to  remove  the  counters  as  the  subtraction 
proceeded.  Those  counters  that  were  left 
expressed  the  remainder.  A third  plan  is 

NUMISMATIC  NOTES 

ben  i.dSm'ff  aoff  bteanbaytowi  9 fxxl  b 
kmweg/blctbai  4 mb  an  ^albs.cSraff 
ewijf  btebnto/m'm  aebt  balb  ^mweg/ 
Waben  4.  (Staff  auff  bte  .vtcrbe/mro 
3 balb  btmveg/  blabs  1 mb  an  t>alba, 
i^lbtt.  3ffl?albtrc, 


g? 3 

P~tr: — 

i®*- 

lllfatfyne  nut  bicfcn  2£jtempeltt 
fytrunten/  2lwdb  alien  anban/fo  biff 
tttrfomctt, 

34<<2— -1731 

*<>14— —145-7 

^ett  4»ff  87^0- —4  360 

^4lbir  p4og  blabs  4704 
75>f2- — -397<> 


n 14- 


-2(^f7 


proba, 

SDupltrbxe  balbiVtcsak^mptbfr 
mberximb  bic  341/tvdcbe  bn  juuor  anff 
gdegt  fyaft/  jb  reefet  ^albxst; 

ttlnlrtf 


Johann  Albert,  Rechenbuchlin,  1534 

From  the  edition  of  1561.  Illustrating  halving 


46 

COMPUTING  JETONS 

the  one  here  shown  in  the  page  from 
Michael  Stifel’s  Deutsche  Arithmetica 
(Niirnberg,  1545,  fol.  4,  v).  The  case  is  the 
subtraction  of  984,392,760  from  9,286,170,- 
534.  The  larger  number  is  set  down  by 
counters  in  the  left-hand  column,  the 
smaller  number  is  written  at  the  left  of  this 
column,  and  the  remainder  appears  at  the 
right.  Stifel  begins  with  the  highest  order, 
changing  92  (hundred  millions)  in  the 
larger  number  to  80+  12.  He  is  then  able 
to  take  9 from  12,  and  his  result  thus  far  is 
83  (hundred  millions),  which  he  represents 
by  counters  in  the  right-hand  column. 
In  a similar  manner  he  proceeds  with  the 
other  orders. 

In  a case  like  that  of  21,346—  7,999,  it 
was  not  unusual  to  arrange  the  larger 
number  so  that  the  subtraction  could  easily 
be  made  without  any  trouble  in  borrowing. 
For  example,  the  Dutch  arithmetician 
Gielis  vander  Hoecke  (Antwerp,  1537) 
places  the  larger  number,  21,346,  in  the 
right-hand  column  as  on  page  48.  He 
then  reduces  this  to  1 ten  thousand  + 5 
thousand  (space)  + 5 thousand  (line)  -f  5 

NUMISMATIC  NOTES 

£Vr(£r(t  tfepf 

$Tu  madden  » fcfjdffei/tm « * fdjdffei  macTjC  i COMter* 
3P  he  fra^/tt>ic  hi  eo  atteo  f om  hinge i 
Sftfacfctalleo  83  Walter/  9 fcfjojfd/  vnD  9 wfc* 
0tel>t  Dife  fumma  atfo  aujf  Den  linien. 


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von  Du  jubfrahren  t vilt)  aujf  Die  (mien  / tmD  Die$al 
oDcrfum/DieDuDaruon  fubtrafnren  miif/Die  mag- 
jhumfmbebalten/oDermagjtftefur  tnclj  fcheibeit 
mit  Dcr  frepben/ oDer  magg  fie  $uriincfen  ftanD  Der  gclcgtm$a( 
fdjmben/nue  Du  filjejf  am  nac^foigenDcn  ejrempei. 


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Michael  Stifel,  Deutsche  Arithmetica , 1545 

Illustrating  subtraction 


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bacrnagr9ptmctt§ein0bcrepbt  flmcbcrfibeopbfe 
im  be  rsiic^te^m  bsrtg 


From  the  Arithmetic  of  Gielis  vander 
Hoecke,  1537 

Illustrating  subtraction 


COMPUTING  JETONS 

49 

hundred  (space)  + 7 hundred  (line)  + 5 
tens  (space)  + 8 tens  (line)  + 10  (space) 
-f  6 (line),  which  he  places  in  the  middle 
column.  He  then  subtracts  7 thousand 
from  the  1 ten  thousand  + 5 thousand 
(space)  + 5 thousand  (line)  and  places  the 
counters  (1  ten  thousand-!-  3 thousand)  in 
the  left-hand  column.  The  rest  of  the  sub- 
traction is  performed  in  a similar  manner, 
the  counters  at  the  left  showing  the  result, 
13,347- 

In  general,  it  was  not  the  custom  to 
devote  much  space  to  explaining  the  opera- 
tions with  the  counters,  this  being  left  to 
the  teacher.  Thus  Hudalrich  Regius 
(Vtrivsque  arithmetices  epitome , Strasburg, 
1536,  but  from  the  Freiburg  edition  of 
1550,  fol.  97,  r)  gives  only  two  examples  in 
subtraction,  and  depends  wholly,  except 
for  a brief  rule,  upon  the  diagrams  given 
on  the  following  page.  The  first  he  calls 
an  “Exemplvm  de  Linea”  and  the  second 
an  “Exemplvm  de  Spacio,”  but  there  is  no 
essential  difference  between  them.  In  each 
case,  if  a simple  subtraction  is  impossible, 
a counter  is  removed  from  the  space  or  line 

NUMISMATIC  NOTES 

EPITOME*  ?7- 

Subtrdhcndiu . Superior.  Rdifluf, 

"• 


4" 


— • 

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EX  EM  PL  VM  PE  SPACIO. 
Subtruhcndui.  Superior . Relifiuf. 


A A 1 At 

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1 p 6 1 

N 


Hudalrich  Regius,  Vtrivsque  arithmetices 
epitome , 1536 

This  reproduction  is  from  the  1550  edition.  Illus- 
trating subtraction 


COMPUTING  JETONS 

51 

above,  and  its  equivalent  is  placed  on  the 
line  (five  counters)  or  in  the  space  (two 
counters)  below. 

In  a case  involving  the  subtraction  of 
denominate  numbers,  computers  often  set 
down  the  different  denominations,  each  in 
its  proper  Cambien.  They  then  subtracted 
by  actually  taking  away  the  counters  as 
required  by  the  problem,  “borrowing”  a 
counter  from  a line  whenever  necessary, 
and  repaying  the  debt  by  placing  two 
counters  in  the  space  below.  The  solution 
on  page  52  is  from  Ein  N ewes  Rechen 
Buchlein  auff  Linien  vn  Federn  (Julius- 
friedenstedt,  1590,  fol.  cij,  r),  by  Eberhard 
Popping,  one  of  the  later  German  arith- 
meticians to  make  use  of  counter-reckon- 
ing. The  problem  is  to  subtract  6324 
florins,  16  groschen,  7 pfennigs,  1 heller 
from  9867  florins,  8 groschen,  3 pfennigs, 
and  only  the  result  is  shown  on  the  count- 
ing table, — this  being  incorrect  in  the 
number  of  groschens. 

For  a simple  illustration  of  the  work  in 
multiplication  the  facsimile  page  from 
Bathasar  Licht  (Leipzig,  c.  1500),  will 

NUMISMATIC  NOTES 

ft 


a 


$ 


f)E 


p •itj 
• 

■ 

A A A A 

• 

# 

3tcm/  gin  ©iaM  3und*r 
3cr!id;cr  aufffutifft  in  gilnff  iEmtttnen 
<mff ju^cbcn / d 4 7 4-  ft  / 18.  / 9-  $ / 

ZiarauffQaterWicr  2 ermine  empfan* 
gen/^fibiegraaf/  33icuicl  manjfcme 
jum  Silnfften  ^ermine  tioc%  ju  <jeb<n 
fdjultng  fen* 
ffacit/ 

1 828.  ft/io.g^/o$j/l.l)V. 
ft  nSS  9 W 

2198  ^ * r 

) 6 4 7 i O 

I 40  2 O d I 

d 8 I 9 8 * 


g&tif  f§me  alfo:  ©ummirbie 

53icr  Hermine  jufammen/mnsM  fdmpe 
mm6s  <ibc  con  ter  ^euptfumma  / ta» 
W<ifon&iltb(r$t»nfftc2>rmin. 

C i|  3‘<m 


Eberhard  Popping,  Ein  Newes  Rechen 
Buchlein  auff  Linien  vn  Federn,  1590 

Illustrating  subtraction 


COMPUTING  JETONS 

53 

serve  the  purpose.  The  book  was  pub- 
lished without  date,  but  the  dedicatory 
epistle  closes  with  the  words  “Vale  ex 
nostra  academia  Lyptzen  Anno.  1500,”  so 
that  the  illustration  represents  the  process 
as  it  must  have  been  performed  in  Leipzig 
at  the  close  of  the  fifteenth  century.  The 
author  recommends  the  learning  of  the 
multiplication  table  up  to  4 X 9.  In  cases 
where  it  was  needed  beyond  this  point, 
the  arithmeticians  of  that  period  had  a 
simple  and  convenient  rule  for  multiplying 
on  the  fingers.  In  the  example  on  page  54 
the  author  first,  on  a preceding  page,  says, 
“Volo  multiplicare  32  cum  43.  Ita  pone.” 
(“I  wish  to  multiply  32  and  43  together. 
Place  the  counters  thus.”)  He  first  places 
the  32  in  the  left  column  as  seen  in  the 
facsimile.  He  then  proceeds  with  his  ex- 
planation substantially  as  follows:  Begin- 
ning with  the  tens,  we  see  that  4 times  3 is 
12;  write  the  1 on  the  thousands’  line 
(which  he  does  not  mark  with  a cross  as  the 
other  writers  usually  did),  and  the  2 on  the 
hundreds’  line;  3 times  3 are  9,  and  this 
being  9 tens  we  place  a counter  above  the 

NUMISMATIC  NOTES 

^fcaltfeUcatfr  *ba  non*  rta  it*  tuer.fcpotm  vtrinffe  00 
mmCricoilegifhYitttcet)  relict  u in  fcmftipltcatu  cfiidncrt 
fdtcto  pcojdart  Stoliter  teer  raffs  ccgnofcae. 
Jftolttplwaridna 

iz  ^nWpUcane  fuodactam 

4$  » |76 


f+7  4* 

V $ 

W Ofipoffte  fpeciei  pioba  in  ^rtmJ  ptorogaf  fpect'2.  &uia  t>| 
tttfio  pjobat  mftiplteatid$*medtatto  buplieattonem.Subtra 
ctio  additionem  ct  econtra. 

B imufiont  two  runt  ofr 

fernandaer  4&ftipticati07  Subtracts  iContn* 
irnrur  autbt'mfio  in  buas  TKegfas.lpama  a lupe* 
rio.nb9  ttfcendercfocipfas  vbttjdenfcuc^  vltttna 
ftimfba'8  tabule  relicti  via  iDultiplicatfs  in  pjoieertlito  t>igf 
to  fubtectis  baberi  potefr  .TCotime  fubtraettoms  mote  a li> 
neatogtto  facts  tllnd^ducru  cetrabaf.  bine  bigitu  trafpone 
pioyt  fequente  lineS.qctenteiteru  in  fequenti  bt  mfotte  flgtl  i 
rd  tuft  ipltcando  qd  pzodnctu  linee  fefcentio  bigiti  suffer* 
iWiincitacefceiidereitcebttfemfeTPIifi  pma  bunions  qefcere 
iuberer.cemu  vbi  beffntfh.fu^bdc  linei  in  cabto  ofpo  foliate 
n*8  portaf  imerus  cftierts  «£t  in  reliquip  iacentibo  pioiectftib) 
ra  liter  biuifiue  feme  pjocedete  opotteat  Donee  ad  infimali> 
tied  puennJ  fuerit  16 ft  tt  in  Piuifione  cauend  u.ne  tilts  mrnie^ 
r’liuensmuemaf.qualealtq  linearS  feques  pafliira  no  eflet 
Qecunda  IRegfa  Singer*  bmidend’btmfozeeft  mine:.  bi 
mTons  medietas  (ft  adeft)abundendo  aufferatnr.et  fubt’li* 
neam  oigito  tacta.m  fpacio  vn  us  ptoiectilis  ponatur.^S  to 
taPinifionefactafioU'qmd  refidui  minus  Piuifote  relmquif 
relictu  appellaf  qd  cuiuMtufoiefractiofieconfliruere  tntelli> 
gifur  iSjemplnm  volo  Dundere  1 $ 7 6 per  4 5 


Balthasar  Licht,  Arithmetic,  c.  1500 

Illustrating  multiplication  and  the  check  of  “casting 
out  nines” 


COMPUTING  JETONS 

55 

tens’  line  and  4 counters  on  the  line;  2 times,. 
4 are  8,  and  this  being  8 tens  we  place  a 
counter  above  the  tens’  line  and  3 counters 
on  the  line.  He  now  readjusts  the  counters 
thus  placed,  carrying  the  2 fifties  to  the 
hundreds’  line,  and  5 tens  to  the  fifty  space. 
He  finally  multiplies  2 by  3 and,  for  the 
product,  places  1 in  the  fives’  space  and 
one  on  the  units’  line.  The  work  then 
appears  as  shown  in  the  facsimile.  The  two 
crosses  below  the  units’  line  indicate  the 
check  by  “casting  out  nines.”  The  rest  of 
the  page  is  given  to  the  first  steps  in 
division. 

Another  example  in  multiplication,  from 
the  Latin  work  of  Joannes  Noviomagus 
(. De  Nvmeris  Libri  77,  Paris,  1539,  but  from 
the  Deventer  edition  of  1551,  fob  Eij,  r ), 
shows  the  operation  of  finding  14  times 
2468  by  the  use  of  the  counters.  The  author 
begins  with  the  lowest  order  and  reduces 
as  he  proceeds,  the  result  being  shown  in 
the  right-hand  column. 

The  operation  of  division  was  always  a 
difficult  one  before  the  Hindu-Arabic 
numerals  became  generally  known  and 

NUMISMATIC  NOTES 

2.IBER  F. 

24^  per  14  multiplied . 


m 

• 

» 

# 

-••• — -- 

— “HI# 

Cbferud  ut  mmmo  pofito  in  Jpdcio  digitus 
tofioceturin  lmcd,eui  facium  ejlfubieftum^t  fub * 
hto  nutnmo  ex  facto,  id  dextrum  ponMurdiuidcn* 
tis  numeri  dimidium,  ut  65 2 per  20.  hue  formant 
fequitur « 


Joannes  Noviomagus,  De  Nvmeris  Libri 

II,  1539 

Illustrating  multiplication 


ARITHMETICS  43 

fto  digito  fecund <e  lines jutmnms  in  fummd  pnftus 
demrium  efjicit  qui  dufcrendut  ponendusq;  iuxia  di * 
gtu.Dcinde  proximo  quinariu  udens,uel  dimidium 
denmj  trdmferendus  ad  fydcium  leuum  fub  digito. 
Ter  tins  jhnili  ratione  tranfyofito  uel  ablato  digi-.o, 
( pnendus  in  infim  lined. 


Joachim  Sterck  van  Ringelbergh, 

Lvcvbrationes,  1541 


Illustrating  two  simple  cases  of  division 


OO 

LO 

COMPUTING  JETONS 

used  in  Europe,  say  in  the  fifteenth 
century.  It  could  be  performed  with  the 
Greek  numerals,  or  even  with  those  used 
by  the  Egyptians  and  other  early  peoples, 
but  it  was  always  looked  upon  as  a process 
to  be  avoided.  The  illustration  on  page  57 
is  from  a work  by  Joachim  Sterck  van 
Ringelbergh  (Opera,  Leyden,  1531,  but  this 
illustration  from  the  Basel  edition  of  his 
Lvcvbrationes , 1541,  p.  415)  and  shows  the 
operation  in  its  simplest  form.  Two  prob- 
lems are  given  on  the  page,  the  first  being 
the  division  of  160  by  10,  in  which  the 
counters  in  the  right-hand  column  are 
merely  lowered  one  line  to  form  the  result 
in  the  left-hand  column;  and  the  second 
being  the  division  of  3500  by  100,  which  is 
performed  in  a similar  fashion.  The  most 
difficult  case  that  Ringelbergh  considers  is 
that  of  600  divided  by  24,  of  which  no 
explanation  is  given,  all  of  which  shows 
how  difficult  the  process  was  considered 
even  in  his  time. 

The  illustration  from  Recorde’s  Ground 
of  Artes  ( c . 1542,  but  from  the  1596  edition) 
gives  an  idea  of  the  method  of  beginning  a 

NUMISMATIC  NOTES 

Diuifion. 


Diuifion, 


Hrtffef  dotone  (be  dioifor,  for 
feare  of  forgetting,  and  then 
fet  the  number  that  Shall  be  di- 
ufoed,  attbe right  fide,  Co  farre 
from  tbe  diuifor,tbat  the  quote# 
entmap  be  fet  bcttoeene  them: 

aaformmpie. 

3f  2iy  (beepecoff  45  f > fobat  did  euerp  (beepe 
eoS?  SCo  kitoto  tm * 3 Should  dtnise  tbe  to&ole 


fornme,  that  i#4?r  bp  but  that  cannot  be: 

therefore  mat  3 frt  reduce  that  45  f into  a 
U&r  denomination*  ad  fntb  (billing*,  then  3 
multiplier  bp*o*and  it  fepooitbat  Cum  (ball 
3 dtoidebptbt  number  offljeepe,  fcbicbte^y, 
t&efetfeo  number*  therefore  3 Ut  tbna* 


SCben  begin  3 at  the  bigbeE  line  oftbediuf- 
Bend, and  fefee  boto  often  3 map  baue  the  Dint- 
for  therein,  and  that  map  3 ooe foure  toned! 
then  far  3 foure  timed  2 are  8,tobich  ffgcafei 
from  p, there  reEetbbut  i$u$t 

8tm 

Robert  Recorde,  Ground  of  Artes,  c,  1542 

This  reproduction  is  from  the  1596  edition.  Illus- 
trating division 


6o 

COMPUTING  JETONS 

practical  problem  in  division.  The  problem 
requires  the  division  of  £45  by  225,  and  the 
facsimile  shows  the  necessity  for  first  reduc- 
ing the  £45  to  900  shillings.  Recorde  then 
takes  4X  200  from  900  and  has  100  left, 
after  which  he  shows  that  the  remaining  25 
of  the  225  is  contained  in  this  remainder 
four  times. 

There  arose  in  early  times,  possibly  in 
India  but  spreading  rapidly  to  the  north 
and  west,  a commercial  rule  which  went  by 
the  name  of  Rule  of  Three.  Its  nature  may 
be  inferred  from  a single  problem  taken, 
with  slight  variation  in  terms,  from  Eyn 
new  kunstlich  behend  vnd  gewiss  Rechen- 
buchlin,  written  by  Henricus  Grammateus, 
or  Heinrich  Schreiber  (Vienna,  1518,  but 
from  the  Frankfort  edition  of  1535,  fol. 
B,  vij,  v)\  “If  4 dreilings  of  wine  cost  90 
florins,  3 schillings,  18  pfennigs,  how  much 
will  7 dreilings  cost?”  Here  three  terms 
are  given,  and  the  rule  was  that  the  fourth 
could  be  found  by  multiplying  the  second 
and  third  together  and  dividing  by  the 
first.  How  this  was  done  in  numerals  is 
shown  in  the  upper  part  of  the  facsimile 

NUMISMATIC  NOTES 

£ * 0 
t 

* 

* 9* 

4 4 ( ; 9 %»  

S*atis8]i/  *$/9V[' 

flDann  bu  at»er  fol4>c  returning  ober  bcr  glei 
$cnir>i{r  mcdben  t>  jf- btr  Imien/f > leg  bie  letft? 
jjI  pjfbie  limen  gegen  ber  Itncfen  &«nb/pnnb 
multiplier  bur4>  «lle  tnun$  in  fimbcr£eif/pnb 
leg  tin  igiicbr  murm  in  Jr  fdb/pri  teyl  burtfc  bie 
ertfejdinaUergetfalttt'icin  bban  e rempeicil 

* fi  gc|d?d?en/d@  bann  |>ie  ivurr  gefe$en. 

24.  ib  7^  tS%  s Relief/  9«ib 

f acit/8491  f{  c 

■ — H — - — % 


•SR — 

■V 

•»* 

9 

•9999 

d 

)k 


l§U  # - 


Henricus  Grammateus,  or  Heinrich 

Schreiber,  Eyn  new  kunsllich  behend 
vnd  gewiss  Rechenbuchlin,  1518 

This  reproduction  is  from  the  1535  edition.  Illus- 
trating the  Rule  of  Three 


62 

COMPUTING  JETONS 

on  page  61,  the  lower  part  showing  how  a 
similar  problem  could  be  solved  by  the  aid 
of  counters. 

A much  more  interesting  illustration  of 
the  use  of  the  counters  in  the  Rule  of 
Three  is  the  one  here  given  from  an  anony- 
mous manuscript  written  at  Salisbury, 
evidently  in  the  Cathedral  School,  in  1533. 
The  interest  lies  chiefly  in  the  fact  that 
manuscripts  on  counter  reckoning,  written 
in  England,  are  very  rare,  and  that  this  one 
is  a particularly  good  piece  of  work.  The 
first  part  of  this  manuscript  relates  to  the 
computus,  that  is,  to  the  computations  of 
the  calendar, — Declaratio  Calendarii  et 
Almanack  huius  Ciste.  The  second  part  is 
entitled  Ars  supputandi  cum  Denar  Us.  The 
whole  is  written  on  vellum  and  is  one  of  the 
most  interesting  of  the  sixteenth-century 
manuscripts  in  Mr.  Plimpton’s  library. 
The  problem  states  that  two  things  cost 
£6,  from  which  it  requires  the  cost  of 
twelve  things.  The  counters  in  the  three 
columns  at  the  left  represent  2,  6,  and  12. 
The  answer,  xxxvi,  is  written,  under 
“Quartus  incognitus,”  in  the  fourth  column. 

NUMISMATIC  NOTES 

wumeru  pev  sccx*ndurnultipltcA^Suff4ejcai 
per. Vi . ci^pu eniunt'4 d.Lxxi  i . cjuc^p  pvtniu 
nurVtettu  dimdc.  suppJc  pa*  duo,ef  cjH4it^ 
mi  m crus  sujj.xxxvi  .pnus  mcogmtus,pre* 
h«  duodecim  Lipidum  efro/kndens.  lot 
bis  omnibus  fin  ifi5,s  ecu  r»d  us  numcius  ef 
oju^rKas^scm^  de  eadem  rc 


.< 

/- 

primus  Tin  f 

7 

V ■ - 

/ 

T#t4iii«  n«rm^ 

f j 

HlCOgUlhlS. 

JLcs  empfa — 

_ Qudlio. 

r~\ 

O 

/“V 

Ly 

xxxvi. 

duo  lapidr*.  Prf.yv  1 ^ $~*^^^***>A=*i**"t — 

From  an  Anonymous  Manuscript  of  1533 

Written  at  Salisbury  England.  It  shows  the  compu- 
tation by  jetons  in  solving  a problem  in  the  Rule  of 
Three 


64 

COMPUTING  JETONS 

HISTORY  OP  MINTED  JETONS 

I have  thus  far  spoken  of  the  rise  of  the 
jeton  in  ancient  times  and  of  its  signifi- 
cance and  use  in  numerical  computation. 
It  remains  to  say  a few  words  concerning 
those  minted  pieces  which  have  come  down 
to  us  from  the  Middle  Ages  and  which 
constitute  the  chief  point  of  contact  with 
the  work  of  the  numismatist.  This  part  of 
the  general  topic  has  been  so  thoroughly 
treated  by  Professor  Barnard,  however, 
that  there  remains  but  little  to  be  done 
except  to  call  attention  once  more  to  his 
great  contribution  to  the  subject.  The  few 
illustrations  which  I give  are  from  speci- 
mens in  my  own  collection,  and  are  in- 
cluded for  the  purpose  of  completing  this 
elementary  presentation  of  the  subject 
rather  than  on  account  of  any  rarity  of  the 
pieces  themselves.  They  represent  such 
ordinary  counters  as  were  prepared  in 
Germany,  chiefly  at  Niirnberg,  for  the  use 
of  computers  in  various  parts  of  Europe. 

Naturally  the  greatest  interest  in  medie- 
val counters  lies  in  the  Italian  pieces,  Italy 
having  been  the  source  from  which  were 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

65 

derived  the  methods  of  computation  used 
in  the  northern  European  countries.  As 
already  stated,  the  use  of  the  abacus  was 
abandoned  there  much  earlier  than  it  was 
north  of  the  Alps.  The  commerce  which 
Venice,  Pisa,  and  Genoa  had  with  the  East 
tended  to  bring  the  Hindu-Arabic  numerals 
into  practical  use  in  Italy  long  before  they 
became  familiar  in  the  less  accessible 
countries  of  France,  England,  Germany, 
and  the  Netherlands.  For  their  early 
computations  the  merchants  may  have 
used  the  Roman  counters, — usually  disks 
of  bone  or  of  baked  clay;  they  may  have 
found  the  grooved  abacus  more  convenient; 
or  they  may  have  used  the  digital  compu- 
tation which  was  international  during  a 
long  period  and  which  is  still  found  in 
Russia,  Poland,  and  certain  of  the  Balkan 
states. 

About  the  year  1200,  however,  the 
Lombard  bankers  and  merchants  began  to 
use  a minted  type  of  counter.  From  that 
time  on  until  about  the  close  of  the 
fourteenth  century  such  counters  seem  to 
have  been  used  in  Italy,  often  in  a half- 

AND  MONOGRAPHS 

1 66 

COMPUTING  JETONS 

' 

hearted  way,  but  in  the  fifteenth  century 
even  this  use  died  out.  The  Treviso 
arithmetic  of  1478,  the  first  work  on  com- 
putation, to  appear  from  the  press,  makes 
no  mention  of  counters,  and  no  other 
Italian  textbook  on  the  subject,  printed  in 
that  century,  discusses  the  matter.  Because 
of  the  fact  that  the  mercantile  and  banking 
class  in  Italy  abandoned  the  use  of  counters 
so  long  before  the  rest  of  Europe,  most  of 
the  extant  specimens  are  confined  to  the 
fourteenth  and  fifteenth  centuries.  Such 
pieces  are  very  rare,  and  the  only  worthy 
description  that  we  have  of  them  is  a recent 
one  by  Professor  Barnard  (“Italian  Jet- 
tons,” Numismatic  Chronicle , vol.  xx  (4), 
for  1920). 

The  counter  of  numismatic  nature  first 
appeared  in  France  about  the  same  time 
that  it  appeared  in  Italy,  that  is,  early  in 
the  thirteenth  century.  The  earliest  identi- 
fied piece  mentioned  by  Professor  Barnard 
is  one  that  seems  to  have  belonged  to  the 
household  of  Blanche  of  Castile  (1200- 
1252),  queen  of  Louis  VIII.  Since  the  use 
of  these  pieces  is  explained  by  Ian  Tren- 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

67 

chant  as  late,  at  least,  as  the  1578  edition 
of  his  Arithmetiqve,  we  may  conclude  that 
numismatic  jetons  were  employed  in 
France  for  ordinary  computation  for  a 
period  of  about  four  hundred  years  (1200- 
1600). 

In  the  sixteenth  and  seventeenth  cen- 
turies the  favored  land  of  the  counter 
(. Rechenpfennig ) was  Germany.  Although 
her  merchants  knew  the  Hindu-Arabic 
numerals  and  could  operate  with  them,  her 
Rechenmeisters  made  common  use  of  the 
abacus  long  after  most  other  countries  of 
Western  Europe  had  virtually  abandoned 
it.  Her  most  popular  arithmetics  of  the 
sixteenth  century  coupled  reckoning  “auff 
Linien”  with  that  by  the  “Feder,”  and 
apparently  her  merchant  apprentices 
favored  the  ancient  method.  The  names  of 
Hans  Schultes,  the  Krauwinckels,  the 
Laufers  (Lauffers),  and  others  appear  on 
thousands  of  extant  jetons  of  Nurnberg 
manufacture,  and  these  pieces  were  sent  to 
all  parts  of  Europe,  being  manufactured 
for  France,  England,  the  Netherlands, 
Austria,  and  the  smaller  states,  as  well  as 

AND  MONOGRAPHS 

68 

COMPUTING  JETONS 

for  those  cities  which  now  belong  to  modern 
Germany.  The  illustrations  on  Plates  I-IV 
are  selected  from  the  Nurnberg  products. 

In  the  Low  Countries,  jetons  were  used 
as  early  as  the  fourteenth  century,  but  the 
computing  pieces  now  commonly  seen  in 
museums  and  the  cabinets  of  numismatists 
are  of  the  fifteenth  and  sixteenth  centuries. 
The  medallic  jetons  of  the  seventeenth 
century  could  scarcely  have  been  generally 
used  for  computing  purposes,  for  the 
arithmetics  of  these  countries  never  paid 
much  attention  to  the  subject,  and  those 
of  the  seventeenth  century  rarely  men- 
tioned it. 

Spain  gave  but  little  attention  to  the  use 
of  the  counters  after  the  invention  of  print- 
ing. Such  jetons  as  she  struck  were  prob- 
ably, in  most  cases,  for  other  purposes  than 
computing.  A few  pieces  were  struck  in 
Portugal  in  the  sixteenth  century,  and 
were  apparently  used  for  computation. 

English  jetons  of  the  fourteenth  century 
are  to  be  seen  in  numismatic  collections, 
but  beginning  about  the  middle  of  the 
century  the  need  was  commonly  met  by 

NUMISMATIC  NOTES 

COMPUTING  JETONS 

69 

pieces  made  abroad, — at  first  by  Flemish 
craftsmen,  but  later  by  those  of  Niirnberg. 
As  already  stated,  counter  reckoning  went 
out  of  use  about  the  close  of  the  sixteenth 
century,  although  jetons  for  gaming  pur- 
poses were  sent  over  from  Germany  until 
well  into  the  eighteenth  century. 

SUMMARY 

The  points  which  I have  endeavored  to 
make  may  be  summarized  briefly  as 
follows: 

1.  The  ancient  notations  were  so  in- 
convenient as  to  render  inevitable  the  use 
of  mechanical  aids. 

2.  These  aids  were  of  various  kinds,  and 
go  back  probably  to  prehistoric  times. 

3.  The  chief  interest  for  mathematicians 
lies  in  the  field  of  computation  and  con- 
cerns the  various  forms  of  the  line  abacus, 
the  methods  employed  in  calculation,  the 
steps  that  slowly  led  to  the  modern  calcu- 
lating machine,  and  the  prospects  of  the 
development  of  simpler  and  less  expensive 
devices  that  will  render  nearly  all  computa- 
tion mechanical. 

AND  MONOGRAPHS 

70 

COMPUTING  JETONS 

4.  The  chief  interest  for  the  historian 
lies  in  a study  of  the  human  needs  which 
the  abacus,  in  its  various  forms,  tended  to 
satisfy,  and  also  in  the  possibility  that 
ingenuity  will,  as  stated  above,  devise  a 
more  efficient  machine  at  a low  cost  so  that 
human  energy  may  be  still  further  con- 
served through  mechanical  calculation. 

5.  The  chief  interest  for  the  numismatist 
lies  not  so  much  in  the  use  of  the  jeton  as  in 
its  history  as  a minted  product.  This  prod- 
uct began  to  appear  in  the  thirteenth 
century  and  ceased  to  meet  any  reasonable 
human  need  in  the  eighteenth.  For  the 
real  lover  of  numismatical  science,  how- 
ever, there  is  always  a deep  interest  in  the 
human  story  involved  in  the  pieces  that  he 
examines,  and  it  is  some  phases  of  this 
human  story  that  I have  endeavored  to  set 
forth  in  this  brief  monograph. 

NUMISMATIC  NOTES 

COMPUTING  JETONS 


Plate  I 


Early  Nurnberg  Jetons 
c.  1450-1500 


A Dutch  Jeton  of  1562 


COMPUTING  JETONS 


Plate  II 


The  Rechenmeister  Type  of  Jetons 

Nurnberg,  c.  1500-1553 


COMPUTING  JETONS 


Plate  III 


Niirnberg  Jeton  by  Hans  Schultes 
c.  1550-1574 


Niirnberg  Jetons  by  Hans  Krauwinckel 

e.  1580-1610 


COMPUTING  JETONS 


Plate  IV 


Niirnberg  Jeton  by  Wolf  Lauffer 

c.  1618-1660.  Intended  for  use  in  France 


Niirnberg  Jetons  by  Conrad  Lauffer 

Intended  for  use  in  England  in  the  time  of  Charles  II, 
for  gaming  purposes 


PUBLICATIONS 


Edward  T.  Newell.  The  Alexandrine  Coinage 
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Edward  T.  Newell.  Myriandros — Alexandria 
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Agnes  Baldwin.  The  Electrum  and  Silver  Coins 
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Albert  R.  Frey.  Dictionary  of  Numismatic 
Names.  1917.  31 1 pages.  $5.00. 

Henry  C.  Miller  and  Hillyer  Ryder.  The  State 

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NUMISMATIC  NOTES  AND 
MONOGRAPHS 


1 Sydney  P.  Noe.  Coin  Hoards.  1921.  47  pages. 

6 plates.  50c. 

2 Edward  T.  Newell.  Octobols  of  Histiaea.  1921. 

25  pages.  2 plates.  50c. 

3 Edward  T.  Newell.  Alexander  Ploards.  Intro- 

duction and  Kyparissia  Hoard.  1921.  21 

pages.  2 plates.  50c. 

4 Howland  Wood.  The  Mexican  Revolutionary 

Coinage  1913-1916.  1921.  44  pages.  26 

plates.  $2.00. 

5 Leonidas  Westervelt.  The  Jenny  Lind  Medals 

and  Tokens.  1921.  25  pages.  9 plates.  50c. 

6 Agnes  Baldwin.  Five  Roman  Gold  Medallions. 

1921.  103  pages.  8 plates.  $1.50. 

7 Sydney  P.  Noe.  Medallic  Work  of  A.  A.  Wein- 

man. 1921.  31  pages.  17  plates.  $1.00. 

8 Gilbert  S.  Perez.  The  Mint  of  the  Philippine 

Islands.  1921.  8 pages.  4 plates.  50c. 

9 David  Eugene  Smith,  LL.D.  Computing  Je- 

tons.  1921.  70  pages.  25  plates.  $1.50. 

10  Edward  T.  Newell.  The  First  Seleucid  Coinage 
of  Tyre.  1921.  (In  press.) 


